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\title{《基础复分析》第5章复积分 - 习题}
\author{CGZ ET AL}

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\begin{enumerate}

%## 《基础复分析》习题五

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\item % 1

设 $\gamma$ 是从 $0$ 到 $1+i$ 的线段, 计算 $\int_{\gamma} x \, dz$.
    

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\item % 2

计算:
- $\displaystyle \int_{|z|=r} x \, dz$
- $\displaystyle \int_{|z|=2} \frac{dz}{z^2 - 1}$
- $\displaystyle \int_{|z|=1} |z-1| \cdot |dz|$
    

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\item % 3

设 $P(z)$ 是多项式. $C$ 为圆周 $|z-a|=R$. 计算
    $$
    \int_C P(z) \, d\bar{z}.
    $$
    

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\item % 4

计算:
- $\displaystyle \int_{|z|=2} \frac{dz}{z^2 + 1}$
- $\displaystyle \int_{|z|=\rho} \frac{|dz|}{|z-a|^2} \quad (|a| \neq \rho)$
- $\displaystyle \int_{|z|=1} \frac{e^z}{z^n} \, dz$
- $\displaystyle \int_{|z|=2} z^n (1-z)^m \, dz$
  
  (提示: 利用 $z\bar{z} = \rho^2$ 以及 $|dz| = -i\rho \, dz/z$.)
    

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\item % 5

设 $\Omega$ 是一个平面区域. 设 $\phi(z,t)$ 作为两个变量的函数对 $z \in \Omega$ 和 $t \in [a,b]$ 连续, 且对固定的 $t$ 在 $\Omega$ 内解析. 试证 $F(z) = \int_a^b \phi(z,t) \, dt$ 解析, 且
    $$
    F'(z) = \int_a^b \frac{\partial \phi(z,t)}{\partial z} \, dt.
    $$
 
(提示: 将 $\phi(z,t)$ 表示为 Cauchy 积分.)
    

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\item % 6

设 $f(z)$ 在包含闭圆盘 $|z| \leq R$ 的一个区域内解析, 且对 $|z| \leq R$, 有 $|f(z)| \leq M$. 求 $|f^{(n)}(z)|$ 在闭圆盘 $|z| \leq \rho < R$ 中的上界.
    

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\item % 7

设 $f(z)$ 在单位圆盘解析, 且 $|f(z)| \leq 1/(1-|z|)$. 求 $|f^{(n)}(0)|$ 的最优估计.
    

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\item % 8

证明一个解析函数在一点的逐阶导数不能满足不等式 $|f^{(n)}(z)| > n! \cdot n^n$.
    

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\item % 9

试证解析函数 $f(z)$ 的一个孤立奇点是可去奇点, 如果 $\operatorname{Re} f(z)$ 或者 $\operatorname{Im} f(z)$ 有上界或者下界.
    

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\item % 10

设 $\Omega$ 是平面上以抛物线 $y^2 = 4(1-x)$ 为边界且包含原点的区域. 求将 $\Omega$ 映为单位圆盘的共形映射.


    

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\item % 11

设 $P(z)$ 为 $n$ 次多项式, 且当 $|z| \leq 1$ 时, $|P(z)| \leq M$. 证明对 $R > 1$, 当 $|z| \leq R$ 时, $|P(z)| \leq MR^n$.
    

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\item % 12

设 $g(z)$ 是非常数解析函数, $f(z)$ 是一个函数, 且 $f \circ g$ 解析. 试证 $f$ 解析.
    

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\item % 13

试证全平面上的解析函数如果以 $\infty$ 为极点, 则一定是多项式.
    

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\item % 14

试证全平面上的解析函数 $f(z)$ 如果满足 $|f(z)| \leq M|z|^n$, 其中 $M > 0$ 为常数, 则 $f(z)$ 是次数不超过 $n$ 的多项式.
    

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\item % 15

试证扩充复平面上的亚纯函数一定是有理函数.
    

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\item % 16

试证 $e^z$, $\sin z$ 以及 $\cos z$ 以 $\infty$ 为本性奇点.
    

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\item % 17

试证解析函数 $f(z)$ 的孤立奇点不是 $e^{f(z)}$ 的极点.
    

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\item % 18

设 $f(z)$ 是全平面上的亚纯函数. 如果对任意点 $w \in \overline{\mathbb{C}}$, $f^{-1}(w)$ 最多包含 $n$ 个点, 试证 $f(z)$ 是次数不超过 $n$ 的有理函数.
    

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\item % 19

如果 $f(z)$ 在原点解析, 且 $f'(0) \neq 0$, 证明存在解析函数 $g(z)$, 使得在原点的一个邻域内, $f(z^n) = f(0) + g(z)^n$.
    

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\item % 20

设 $f(z)$ 是单位圆盘内的解析函数, 满足 $|f(z)| < 1$. 证明:
    $$
    \frac{|f'(z)|}{1 - |f(z)|^2} \leq \frac{1}{1 - |z|^2}.
    $$

等号在一点成立, 当且仅当 $f(z)$ 是分式线性变换.
    

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\item % 21

设 $f(z)$ 是上半平面内的解析函数, 满足 $\operatorname{Im} f(z) > 0$. 证明:
    $$
    \frac{|f(z) - f(z_0)|}{|f(z) - \overline{f(z_0)}|} \leq \frac{|z - z_0|}{|z - \overline{z_0}|}, \quad \frac{|f'(z)|}{\operatorname{Im} f(z)} \leq \frac{1}{\operatorname{Im} z}.
    $$

等号在一点成立, 当且仅当 $f(z)$ 是分式线性变换.




\end{enumerate}

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